Pre-Algebra Examples

$\sqrt{x - 13} = 13 - \sqrt{x}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x - 13}}^{2} = {(13 - \sqrt{x})}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 13}$ as ${(x - 13)}^{\frac{1}{2}}$ .

${({(x - 13)}^{\frac{1}{2}})}^{2} = {(13 - \sqrt{x})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x - 13)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x - 13)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 13)}^{\frac{1}{2} \cdot 2} = {(13 - \sqrt{x})}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x - 13)}^{\frac{1}{2} \cdot 2} = {(13 - \sqrt{x})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x - 13)}^{1} = {(13 - \sqrt{x})}^{2}$

Step 2.2.1.2

Simplify.

$x - 13 = {(13 - \sqrt{x})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(13 - \sqrt{x})}^{2}$ .

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Step 2.3.1.1

Rewrite ${(13 - \sqrt{x})}^{2}$ as $(13 - \sqrt{x}) (13 - \sqrt{x})$ .

$x - 13 = (13 - \sqrt{x}) (13 - \sqrt{x})$

Step 2.3.1.2

Expand $(13 - \sqrt{x}) (13 - \sqrt{x})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$x - 13 = 13 (13 - \sqrt{x}) - \sqrt{x} (13 - \sqrt{x})$

Step 2.3.1.2.2

Apply the distributive property.

$x - 13 = 13 \cdot 13 + 13 (- \sqrt{x}) - \sqrt{x} (13 - \sqrt{x})$

Step 2.3.1.2.3

Apply the distributive property.

$x - 13 = 13 \cdot 13 + 13 (- \sqrt{x}) - \sqrt{x} \cdot 13 - \sqrt{x} (- \sqrt{x})$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $13$ by $13$ .

$x - 13 = 169 + 13 (- \sqrt{x}) - \sqrt{x} \cdot 13 - \sqrt{x} (- \sqrt{x})$

Step 2.3.1.3.1.2

Multiply $- 1$ by $13$ .

$x - 13 = 169 - 13 \sqrt{x} - \sqrt{x} \cdot 13 - \sqrt{x} (- \sqrt{x})$

Step 2.3.1.3.1.3

Multiply $13$ by $- 1$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} - \sqrt{x} (- \sqrt{x})$

Step 2.3.1.3.1.4

Multiply $- \sqrt{x} (- \sqrt{x})$ .

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Step 2.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + 1 \sqrt{x} \sqrt{x}$

Step 2.3.1.3.1.4.2

Multiply $\sqrt{x}$ by $1$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + \sqrt{x} \sqrt{x}$

Step 2.3.1.3.1.4.3

Raise $\sqrt{x}$ to the power of $1$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

Step 2.3.1.3.1.4.4

Raise $\sqrt{x}$ to the power of $1$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

Step 2.3.1.3.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Step 2.3.1.3.1.4.6

Add $1$ and $1$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + {\sqrt{x}}^{2}$

Step 2.3.1.3.1.5

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 2.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + x^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.5.4.1

Cancel the common factor.

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + x^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4.2

Rewrite the expression.

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + x^{1}$

Step 2.3.1.3.1.5.5

Simplify.

$x - 13 = 169 - 13 \sqrt{x} - 13 \sqrt{x} + x$

Step 2.3.1.3.2

Subtract $13 \sqrt{x}$ from $- 13 \sqrt{x}$ .

$x - 13 = 169 - 26 \sqrt{x} + x$

Step 3

Solve for $- 26 \sqrt{x}$ .

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Step 3.1

Rewrite the equation as $169 - 26 \sqrt{x} + x = x - 13$ .

$169 - 26 \sqrt{x} + x = x - 13$

Step 3.2

Move all terms not containing $- 26 \sqrt{x}$ to the right side of the equation.

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Step 3.2.1

Subtract $169$ from both sides of the equation.

$- 26 \sqrt{x} + x = x - 13 - 169$

Step 3.2.2

Subtract $x$ from both sides of the equation.

$- 26 \sqrt{x} = x - 13 - 169 - x$

Step 3.2.3

Combine the opposite terms in $x - 13 - 169 - x$ .

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Step 3.2.3.1

Subtract $x$ from $x$ .

$- 26 \sqrt{x} = 0 - 13 - 169$

Step 3.2.3.2

Subtract $13$ from $0$ .

$- 26 \sqrt{x} = - 13 - 169$

Step 3.2.4

Subtract $169$ from $- 13$ .

$- 26 \sqrt{x} = - 182$

Step 4

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 26 \sqrt{x})}^{2} = {(- 182)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(- 26 x^{\frac{1}{2}})}^{2} = {(- 182)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(- 26 x^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $- 26 x^{\frac{1}{2}}$ .

${(- 26)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 182)}^{2}$

Step 5.2.1.2

Raise $- 26$ to the power of $2$ .

$676 {(x^{\frac{1}{2}})}^{2} = {(- 182)}^{2}$

Step 5.2.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$676 x^{\frac{1}{2} \cdot 2} = {(- 182)}^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$676 x^{\frac{1}{2} \cdot 2} = {(- 182)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$676 x^{1} = {(- 182)}^{2}$

Step 5.2.1.4

Simplify.

$676 x = {(- 182)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Raise $- 182$ to the power of $2$ .

$676 x = 33124$

Step 6

Divide each term in $676 x = 33124$ by $676$ and simplify.

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Step 6.1

Divide each term in $676 x = 33124$ by $676$ .

$\frac{676 x}{676} = \frac{33124}{676}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $676$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{676 x}{676} = \frac{33124}{676}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{33124}{676}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $33124$ by $676$ .

$x = 49$